ON CONFINED MOTION. 35 



slant, at different points of the curve, and observing that they meet at the 

 lowest point. (Plate II. Fig. 24.) 



The absolute time of the descent or ascent of a pendulum, in a cycloid, 

 is to the time in which any heavy body would fall through one half of the 

 length of the thread, as half the circumference of a circle to its diameter.* 

 It is, therefore, nearly equal to the time required for the descent of a 

 body through -| of the length of the thread ; and if we suffer the pendulum 

 to descend, at the same moment that a body falls from a point elevated 

 one fourth of the length of the thread above the point of suspension, this 

 body will meet the pendulum at the lowest point of its vibration. (Plate 

 II. Fig. 24.) 



Hence it may readily be inferred, that since the times of falling through 

 any spaces are as the square roots of those spaces, the times of vibration 

 of different pendulums are as the square roots of their lengths. Thus, the 

 times of vibration of pendulums of 1 foot and 4 foot in length, will be as 

 1 to 2 : the time of vibration of a pendulum 39-rW inches in length is one 

 second ; the length of a pendulum vibrating in two seconds must be four 

 times as great. 



The velocity with which a pendulous body moves, at each point of a 

 cycloidal curve, may be represented, by supposing another pendulum to 

 revolve uniformly in a circle, setting out from the lowest point, at the same 

 time that the first pendulum begins to move, and completing its revolution 

 in the time of two vibrations ; then the height, acquired by the pendulum 

 revolving equably, will always be equal to the space described by the 

 pendulum vibrating in the cycloid. (Plate II. Fig. 24.) 



It may also be shown, that if the pendulum vibrate through the whole 

 curve, it will everywhere move with the same velocity as the point of the 

 circle which is supposed to have originally described the cycloid, pro- 

 vided that the circle roll onwards with an equable motion. 



All these properties depend on this circumstance, that the relative force, 

 urging the body to descend along the curve, is always proportional to the 

 distance from the lowest point ; and it happens in many other instances of 

 the action of various forces, that a similar law prevails : in all such cases, 

 the vibrations are isochronous, and the space described corresponds to the 

 versed sine of a circular are increasing uniformly, that is to the height of 

 any point of a wheel revolving uniformly on its axis, or rolling uniformly 

 on a horizontal plane. 



The cycloid is the curve in which a body may descend in the shortest 

 possible time, from a given point to another obliquely below it.t It may 

 easily be shown that a body descends more rapidly in a cycloid than in the 

 right line joining the two points. This property is of little practical 

 utility ; the proposition was formerly considered as somewhat difficult to 

 be demonstrated, but of late, from the invention of new modes of calcula- 



, * Huyg. Horol. Oscil. Part II. Prop. 25. 



t Jo. Bernoulli, Acta Erudit. Lips. 1696, p. 269. Ja. Bernoulli, ibid. 1697, p. 

 211, and Opera, ii. 768. Euler, Acta Petrop. 1733, &c. &c. Lagrange, Miscella- 

 nea Taurinensia, vols. i. and ii. Consult Woodhouse's Isoperimetrical Problems, 

 Camb. 1810 ; or the article Variations in the Encyc. Brit. 



D 2 



